AP Precalculus Section 3.6 Example: The Ferris Wheel Sinusoidal Word Problem

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

Let's use the formula \(y = a \sin(b(x - c)) + d\) to model the height of a passenger on a Ferris wheel. Here's a breakdown:

1. **Amplitude (\(a\)):**
- The amplitude is the vertical distance from the center to the highest point on the Ferris wheel. It is also the radius of the Ferris wheel. Let's say \(a = 10\) meters.

2. **Period (\(T\)):**
- The period is the time it takes for one complete revolution of the Ferris wheel. Let's say it takes 60 seconds for the Ferris wheel to complete one revolution. Therefore, \(T = 60\) seconds.

3. **Phase Shift (\(c\)):**
- If the Ferris wheel starts at its highest point, there is no phase shift (\(c = 0\)).

4. **Vertical Shift (\(d\)):**
- If the Ferris wheel is centered at ground level, there is no vertical shift (\(d = 0\)).

Now, the specific sinusoidal function for the Ferris wheel is:

\[ y(t) = 10 \sin\left(\frac{2\pi}{60} t\right) \]

This function models the height of a passenger at time \(t\) as the Ferris wheel rotates.

To predict the height at specific time values, simply plug in those values for \(t\) and calculate the corresponding heights. For example, to find the height at \(t = 15\) seconds:

\[ y(15) = 10 \sin\left(\frac{2\pi}{60} \times 15\right) \]

Evaluate this expression to get the predicted height at \(t = 15\) seconds. Repeat this process for any other specific time values you're interested in.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#APPrecalculus
#PreCalcProblems
#MathMinds
#AdvancedPreCalc
#TrigTales
#PrecalcPuzzles
#FunctionFiesta
#GraphGoals
#CalcReady
#PreCalcLife
#AlgebraicAdventures
#DerivativeDreams
#IntegrationInsights
#MathematicsMagic
#PreCalcReview
#PrecalcConcepts
#LogarithmLove
#TrigonometryTips
#MathMastermind
#APCalcPrep
#Mathematics
#MathMinds
#Math
#Maths